Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.util.Scanner;

public class Main {

	static int coinCount;
	static double[] coinProbabilityArray;
	static double[][] dp;

	public static void main(String[] args) {

		Scanner scan = new Scanner(System.in);

		coinCount = scan.nextInt();
		coinProbabilityArray = new double[coinCount];

		for (int i = 0; i < coinCount; i++) {

			coinProbabilityArray[i] = scan.nextDouble();
		}

		dp = new double[coinCount + 1][coinCount + 1];

		for (int i = 0; i < coinCount + 1; i++) {

			for (int j = 0; j < coinCount + 1; j++) {

				dp[i][j] = dpMethod(i, j);
			}
		}

		double answer = 0;

		for (int i = coinCount; coinCount / 2 < i; i--) {

			answer += dp[i][coinCount - i];
		}

		println(answer);

	}

	public static double dpMethod(int i, int j) {

		if (coinCount - 1 < i + j - 1) {
			return 0;
		}

		if (i == 0 && j == 0) {

			return 1;
		}

		if (i == 0) {

			return dp[i][j - 1] * back(coinProbabilityArray[i + j - 1]);

		} else if (j == 0) {

			return dp[i - 1][j] * coinProbabilityArray[i + j - 1];

		} else {

			return dp[i - 1][j] * coinProbabilityArray[i + j - 1]
					+ dp[i][j - 1] * back(coinProbabilityArray[i + j - 1]);
		}
	}

	public static double back(double coinProbability) {
		return 1 - coinProbability;
	}

	public static void println(Object target) {
		System.out.println(target);
	}
}